Let $F$ be a field and let $A$ be a unital associative $F$-algebra of dimension $n$. My notes define the group scheme $GL_1(A)$ to be the corepresentable functor $h^A$ corepresented by the hopf-algebra $S(A^*)[1/N]$ where $N:A\to F$ is the norm map considered as an element of $S^n(A^*)$.
I have two questions. Firstly, what is $A^*$ for an algebra? They just mean to consider $A$ as a $F$-vector space, and consequently they take the standard dual-space for this vector space. Secondly, does every unital associative $F$-algebra of dimension $n$ possess a norm? I know of norms arising in the context of composition algebras, like $\Bbb Q(\sqrt{d})$ for $d$ square-free, or the quaternions etc, but how does this arise in general?